Economics Assignment

Q1. Consider the growth model (following Section 3.2 of the Romer textbook) where knowledge evolves according to the equations

A^·(t) = B(a_LL(t))^γA(t)^θ and L^·(t)/L(t) = n.

Like usual, B, γ, n and θ are positive. 0 < a_L < 1. A(0) > 1 and L(0) > 1. Consider (one at a time) the effects of each of the following parameter shifts:

(a) Starting from the balanced growth path at time 0 with θ < 1, an immediate increase in the population growth rate (from n₀ to n₁).

(b) Starting from the balanced growth path at time 0 with θ < 1, an immediate increase in θ to 1.

For each case, consider g_A(t) ≡ A^·(t)/A(t), and g_gA(t) ≡ g·_A(t)/g_A(t). For both g_A and g_gA, (i) specify the value of each immediately after the parameter shift, (ii) note whether or not there has been a discrete change immediately after the parameter shift, and if so, specify the sign of the change (iii) note if either is constant in the long run, and if so, give their new values, (iv) compare the new long run outcome to the outcome on the balanced growth path prior to the parameters shifts.

Q2. Consider the P. Romer model (following section 3.5 of the D. Romer textbook). Suppose there is initially a positive growth rate in output, and then there is a discrete, instantaneous increase in B (the R&D technology productivity parameter). Explain how the increase in B impacts each of the following terms. (Further instructions: for every part, note whether there is an increase or decrease. Provide one mathematical formula that confirms your answer, and explain in a sentence or two of economic language why the increase in R&D productivity leads to an increase or decrease. To simplify the exercise, you do not need to explain why LA increases after the change in B. You may also refer back to your previous explanations in subsequent answers.)

(a) The fixed cost paid by a firm to produce a new idea

(b) The growth rate in the stock of knowledge

(c) The growth rate in output

(d) The growth rate in wages

(e) The growth rate in consumption

(f) The level of the real interest rate

(g) The growth rate in profits from a new idea

(h) The present discounted value of profits from a new idea

Q3. Consider the firm investment model (from Section 9.2 and lecture) but without adjustment costs (so C(I) = 0 always).

(a) What are the first-order conditions for investment and the capital stock under this model? (use the continuous-time version, and simplify as much as possible)

(b) Suppose in this model that there is a discrete change in the interest rate (e.g. the Federal Reserve cuts rates). What happens to K_t and q_t instantaneously?

(c) What does your answer to the previous part imply about how investment responds to interest rate cuts when there are no adjustment costs?

(d) Describe two plausible micro-foundations for why adjustment costs might exist.

Q4. Consider the firm investment model (from section 9.2 and lecture).

(a) Using a phase diagram, draw and describe what happens when there is a permanent increase in the interest rate. Explain what happens to q, K, and I, instantaneously, at the final steady state, and in between.

(b) Answer the same question for a temporary increase in the interest rate.